1. Announcements 11/14 Today: 9.6, 9.8 Friday: 10A – 10E Monday: 10F – 10H 9.6 Only do differential cross-section See problem 7.7 to do most of the work for us

2. Weak Interactions First discovered in  -decay Energy spectrum of electron: Neutrinos Must be some particle carrying off the rest of the energy We now know it is neutron decay Probably an invisible, neutral particle Must be a fermion to conserve angular momentum Must be neutral Must be very light (< 1 eV or so)

3. Other Neutrino Interactions Muon – like a heavy electron, but unstable Decay requires two invisible particles Pion – strongly interacting particle Easily produced in proton-nucleus collisions Decays to muon plus neutrino Neutrinos can be converted back to the corresponding charged particles Electron neutrinos make electrons, muon neutrinos make muons

4. The Leptons We have already discussed the quarks – there are six of them They each come in three colors, and have strong interactions There are six other spin ½ fermions in the standard model, called leptons There are anti-particles for each of these as well Neutrino physics is currently evolving better names are 1 2 3 When weak interactions were first researched, quarks weren’t known We will focus on leptons first

5. Fermi Theory of Muon Decay First attempt – Fermi Theory – assumed this was a basic matrix element Original guess was something like this: New fundamental constant: This is not the only form that respects Lorentz invariance: These five combinations (and linear combinations) are only objects that respect Lorentz invariance including P, T, and PT No combination fit all the available date

6. V – A Theory of Muon Decay In 1956, it was proposed that parity might be violated in weak interactions In early 1957, this was quickly experimentally verified Suddenly there were other possibilities: A   coupling is called a vector coupling, and a    5 is called axial vector We will call this the V – A theory of weak interactions Only left-handed fields participate in these interactions

7. Muon decay rate calculation Treat electron as massless: The  terms are anti-symmetric under  , the other terms are symmetric The cross terms will automatically vanish

8. Muon decay rate calculation (2) This amplitude squared was solved in problem 4.11 Note decay rate rises rapidly as mass increases Weak interactions get stronger as you go up in energy Eventually, get probabilities >1  no good

9. Announcements 11/16 Today:10A – 10E Monday: 10F – 10H Monday:

10. The W - particle This interaction is not renormalizable, since G F ~ GeV -2 Maybe this is not really what is going on? To get this to work, we need W coupling something like: The factor of 2  2 is for convenience later The index  implies the W particle must have polarization vector, like a photon Spin 1, like a photon The W must be charged, unlike a photon The W must be massive, or it would have already been discovered

11. Dealing With Spin-1 Massive Particles Polarization vector satisfies same equations as before: But this time there are three such polarizations For example, if Then the three polarizations are: We need to find For propagator For summing over initial/final states The propagator:

12. Questions from the Reading Quiz “I have no idea what's going on with the groups and the electroweak coupling/interaction. I understand that the SU(2) and U(1) aren't really independent. But it's all confusing and it's making my head hurt.” Spin 1 particles run into trouble with renormalization unless they are gauge- type couplings What we think is going on so far is:

13. A Toy Model – The Two Photon Model Surprisingly, it is sometimes ambiguous which are the actual particles Consider the following toy model: The Carlson two-photon model Classically, if you shake a particle with both types of charge, you would make both types of fields Quantum mechanically, you would create states that are superpositions of each type of field Unless there is something logically picking out particular directions in A 1 A 2 -space, it is not obvious which ones you want to think of as the “real” fields.

14. Rotating Fields Arbitrarily We can change the fields in any arbitrary way, for example We can just as easily work with these fields

15. Announcements 11/16 Today:10F – 10H Monday: 10.1, 10.3 Wednesday: 10.4, 10.5, 10.8

16. Weak Interactions with One Lepton Pair The left- and right-handed pieces of the massive electron have different weak interactions, and should be divided Weak interactions connect the left-handed neutrino and electron Naively, there is one charged lepton field and one neutrino, Without mass, only the left-handed neutrino has weak interactions There is no reason to even believe there is a right-handed neutrino Masses connect the left- and right-handed electrons

17. Mass and Couplings with One Lepton The Feynman rule for W- coupling for one lepton: The Vertex There is no reason, in principle, that the mass can’t be apparently complex This can easily be fixed, for example, by redefining the field e R by a phase: Hence the phase is irrelevant We work with e L and e R ’, and drop the primes

18. Weak Interactions with Multiple Leptons The left- and right-handed pieces of the massive electron have different weak interactions, and should be divided Weak interactions connect the left-handed neutrino and lepton Naively, there are three charged lepton fields and neutrinos, Without mass, only the left-handed neutrino has weak interactions There is no reason to even believe there is a right-handed neutrino Masses connect the left- and right-handed leptons

19. The Weak coupling with Many leptons The Feynman rule for W- coupling for many leptons: The Vertex

20. Complicated mass? There is no reason, in principle, that the mass can’t be apparently complicated This matrix is completely arbitrary We can nonetheless always “change basis” to straighten it out For example, suppose the mass matrix looked like this: Define new states: The new mass matrix is then:

21. Complicated Couplings? We originally had But we now defined new states This makes our W-couplings complicated In the leptons, this can be fixed simply by similarly redefining the neutrinos Drop the irrelevant primes The Vertex

22. Weak Interactions with One Quark Pair The left- and right-handed pieces of the massive quarks have different weak interactions, and should be divided Weak interactions connect the up and down quarks Naively, there is one up quark and one down quark All four of these exist in the Standard Model Masses connect the left- and right-handed quarks

23. The Weak coupling with many quarks The Feynman rule for W- coupling for many quarks: The Vertex Warning: This is actually incorrect! This is different from the leptons As I will explain soon (I hope)

24. Complicated mass? There is no reason, in principle, that the masses can’t be apparently complicated These matrices are completely arbitrary We can nonetheless always “change basis” to straighten them out For example, suppose the mass matrices looked like this: Define new states: The new mass matrix is then:

25. Complicated Couplings? We originally had But we now defined new states This makes our W-couplings complicated In the quarkss, can this can fixed simply by similarly redefining the up quarks? No! This messes up the mass matrix M The couplings really are complicated in the quark sector

26. The Weak coupling with many quarks By appropriate redefinition of the various fields, the mass matrices for the up- and down-type quarks can always be made diagonal and real Such a redefinition will, however, introduce a unitary matrix V into the charged current interactions This matrix is called the Cabibbo-Kobayashi-Maskawa matrix, or CKM matrix Some, but not all, of the parameters of V can be eliminated by appropriate redefinition of the corresponding fields.